Porism

A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem.In modern usage, a porism is a relation that holds for an infinite range of values but only if a certain condition is assumed, for example Steiner's porism.The term originates from three books of Euclid with porism, that have been lost.Note that a proposition may not have been proven, so a porism may not be a theorem, or for that matter, it may not be true.'Porisma est propositio in qua proponitur demonstrare rem aliquam, vel plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quae ad ea quae data sunt eandem habent rationem, convenire ostendendum est affectionem quandam communem in propositione descriptam. Porisma etiam in forma problematis enuntiari potest, si nimirum ex quibus data demonstranda sunt, invenienda proponantur.' A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct result of a proof, analogous to how a corollary refers to a direct result of a theorem.In modern usage, a porism is a relation that holds for an infinite range of values but only if a certain condition is assumed, for example Steiner's porism.The term originates from three books of Euclid with porism, that have been lost.Note that a proposition may not have been proven, so a porism may not be a theorem, or for that matter, it may not be true. The treatise which has given rise to this subject is the Porisms of Euclid, the author of the Elements. As much as is known of this lost treatise is due to the Collection of Pappus of Alexandria, who mentions it along with other geometrical treatises, and gives a number of lemmas necessary for understanding it. Pappus states: Pappus goes on to say that this last definition was changed by certain later geometers, who defined a porism on the ground of an accidental characteristic as'τὸ λεῖπον ὑποθέσει τοπικοῦ θεωρήματος' (to leîpon hypothései topikoû theōrḗmatos), that which falls short of a locus-theorem by a (or in its) hypothesis. Proclus points out that the word porism was used in two senses. One sense is that of 'corollary', as a result unsought, as it were, but seen to follow from a theorem. On the porism in the other sense he adds nothing to the definition of 'the older geometers' except to say that the finding of the center of a circle and the finding of the greatest common measure are porisms. Pappus gave a complete enunciation of a porism derived from Euclid, and an extension of it to a more general case. This porism, expressed in modern language, asserts the following: Given four straight lines of which three turn about the points in which they meet the fourth, if two of the points of intersection of these lines lie each on a fixed straight line, the remaining point of intersection will also lie on another straight line. The general enunciation applies to any number of straight lines, say n + 1, of which n can turn about as many points fixed on the (n + 1)th. These n straight lines cut, two and two, in 1/2n(n − 1) points, 1/2n(n − 1) being a triangular number whose side is n − 1. If, then, they are made to turn about the n fixed points so that any n − 1 of their 1/2n(n − 1) points of intersection, chosen subject to a certain limitation, lie on n − 1 given fixed straight lines, then each of the remaining points of intersection, 1/2n(n − 1)(n − 2) in number, describes a straight line. Pappus gives also a complete enunciation of one porism of the first book of Euclid's treatise.